The Dutch were trailing 1-0 at the break on Friday but by playing their best football of the campaign so far they managed to turn the tide and come back strongly to win 2-1.
"At half time, we said to each other we had to improve things, we had to put more pressure on the Brazilian defence," Sneijder, who scored the winner, told a news conference.
"For 45 minutes we went full throttle and we were rewarded for it. When you score two goals and you win, it's wonderful," he added after the match at Nelson Mandela Bay Stadium.
It was Netherlands' first victory over the South American side since they beat Brazil 2-0 in a World Cup second round group match at the 1974 finals in West Germany.
Sneijder played in five of their eight qualifiers without scoring but has netted three times already at these finals including Friday's 68th minute winner.
He also supplied the cross that was flicked into his own net by Brazil's Felipe Melo to level the scores in the 53rd.
"The feeling is fantastic. If you can eliminate Brazil, one of the biggest, best teams of this championship... If you do that, the feeling is relief," he added.
"Brazil have a very strong defence but nevertheless we scored two goals."
"It was my first headed goal and I don't think it will happen again... it slipped off my bald head and went into the goal," a somewhat surprised Sneijder said.
"We have to take a moment to savour it and from tomorrow, we have to concentrate on the semi-final," he added ahead of their next match on Tuesday at Green Point Stadium in Cape Town.
